Yes, copy editors tighten text. Yes, we cut superfluous words. But whoever decided that "that" is never necessary should be flogged with the OED.
The easy rule: You need "that" if a time element comes between the verb and the dependent clause. An IMF official said Saturday that Cuba was in trouble. You'll see this a lot in ledes, and it helps readers distinguish between the official's saying it Saturday and Cuba's being in trouble Saturday.
The tougher rule (that I may just be making up but makes tons of sense to me and hasn't done me wrong yet): You often need "that" when it follows a verb that can be both transitive and intransitive. She added THAT her team was asked to recommend a change. You don't want the "her team" to be read, even momentarily, as the direct object. "That" fixes the problem nicely.
I acknowledged THAT problems exist with my Web site. This reads so much better than I acknowledged problems exist with my Web site.